The double empathy problem is useful to read about - basically, autistic people typically struggle with neurotypical social expectations. NT communication involves a lot of assumption, subtext, context, implication, missing information which you're expected to fill in (because it's "obvious") etc, usually because it is considered rude to make a direct request, or because giving the "obvious" information implies that the listener is stupid, unfamiliar with the topic etc and may insult them or result in an explanation which is too long. (As I write this not in the correct order, I realise the irony of me explaining that ND explanations can be too long!)
Autistic people often miss a lot of this, or don't know what assumptions to make or what information to fill in, not because they are stupid or unfamiliar with the topic, but because it can be difficult to assimilate parts of things out of context. They can take people overly literally. There is also a lot of (in NT communication) what I tend to think of as social code, which is non-literal, and where you're adding little extra things to interactions in order to communicate an underlying piece of info about your feeling or the relationship between you - it might be something that means "this was a lot of effort, please appreciate it" or "I don't want to overstep boundaries" or "we are just the same, you and me" or "remember, I'm the one in charge" or "I want this interaction to end now/soon".
Because in real life communication, those will rarely be literal but things like self-deprecating humour, or references to pop culture, or even total inversions of the literal meaning (oh it's no trouble at all!) or sometimes nonverbal e.g. tone or body language, they can be difficult to pick up on for autistic people who often prefer a more direct communication style, and may struggle to make sense of something taken out of context (e.g. a pop culture reference). And/or they may come across as pointless (talk about the weather) so may be ignored, or alternately they may be taken much too seriously where a NT person would recognise that it's just social lubricant and not pay attention to the words themselves but note the meaning, usually subconsciously.
This kind of thing led to a medical belief for years that autistic people "don't have empathy" simply because they don't respond in an expected way to NT social cues and/or don't pick up on subtle clues that the other person might be bored, irritated, upset, uncomfortable etc.
This is a very specific, NT-coded model of empathy. So it's incorrect to say autistic people don't have empathy, just because they don't automatically pick up on and perform the social rituals that we tend to code as empathetic.
So OTOH, you could turn this around and say NT people struggle to empathise with autistic people and/or struggle with autistic communication style. NT people use unclear communication and don't get straight to the point or explain what they mean. They may get annoyed at questions they feel are intrusive or unnecessary. They may be unaware of signs that an autistic person is becoming uncomfortable, anxious, or confused about what is being communicated to them. But nobody suggests that NT people don't have empathy.
Autistic people tend to communicate much more easily with other autistic people, because they don't usually do all of that social coding stuff unless they have picked up on the fact that other people do it and repeat it almost rote - it often feels a bit out of place to NTs, usually because they have picked up on a rough context where this happens but not the subtleties of exactly when. Autistic people also often give a thorough and exact explanation, which can come across as too specific or too long to NT people, but is usually welcomed by other autistic people. And they may be more likely to be direct in requests, which again is clearer to other autistic people and usually not taken offensively. Clarifying questions are also welcomed in autistic communication, whereas autistic to NT clarifying questions are sometimes mistaken for being one of the "socially coded" type interactions and taken as a dig at the original person's instructions/authority or seen as a way that the questioning person is trying to get out of following the instruction.
When NT people communicate with other NT people they can usually follow the other person's social codes and context unless there is a barrier such as differing cultural norms (because social codes are extremely culturally specific) - culture shock is a great example to look at this, because for example if you take two cultures which seem similar but the communication norms are quite different, e.g. British/American - the first time one interacts with the other, you get the American wondering if they have upset or offended the Brit, or assuming the Brit is a nervous kind of person because a Brit will often make a lot of self-deprecating jokes, this is our way of demonstrating respect, by reassuring the other person that I don't think I'm better than you. And our level of friendliness towards acquaintances is much more understated. Then the other way around, the Brit may wonder if the American is trying to manipulate or trick them, or think that they are an arrogant sort of person, because the level of friendliness Americans tend to display towards acquaintances is much more obvious and can come across as fake or disengenious to Brits, and because in American culture, confidence communicates competence and trustworthiness, whereas in British culture confidence can look like superiority. Once you get used to this, you can then "read" the person more easily.
The double empathy problem is about how autistic people and NT people fail to understand each other properly, though understand each other much better. And the fact that in these interactions, it is only the autistic person who is accused of not having empathy and expected to change the way they communicate in order that NT people won't misunderstand them.